## Unstable Poles--Unit Circle Viewpoint

We saw in §8.4 that an LTI filter is stable if and only
if all of its poles are strictly inside the unit circle () in
the complex plane. In particular, a pole outside the unit
circle () gives rise to an impulse-response component
proportional to which grows exponentially over time . We
also saw in §6.2 that the *z* transform of a growing exponential does
not not converge on the unit circle in the plane. However, this
was the case for a *causal* exponential , where
is the unit-step function (which switches from 0 to 1 at time 0). If
the same exponential is instead *anticausal*, *i.e.*, of the form
, then, as we'll see in this section, its *z* transform does exist on
the unit circle, and the pole is in exactly the same place as in the
causal case. Therefore,to unambiguously invert a *z* transform, we must know
its *region of convergence*. The critical question is whether
the region of convergence includes the unit circle: If it does, then
each pole outside the unit circle corresponds to an anticausal, finite
energy, exponential, while each pole inside corresponds to the usual
causal decaying exponential.

### Geometric Series

The essence of the situation can be illustrated using a simple geometric series. Let be any real (or complex) number. Then we have

*or*, an anticausal geometric series in (negative) powers of .

### One-Pole Transfer Functions

We can apply the same analysis to a one-pole transfer function. Let denote any real or complex number:

*z*transform is then the causal decaying sampled exponential

Now consider the rewritten case:

where the inverse *z* transform is the inverse *bilateral* *z* transform. In this
case, the convergence criterion is
, or , and
this region includes the unit circle when .

In summary, when the region-of-convergence of the *z* transform is assumed to
include the unit circle of the plane, poles inside the unit circle
correspond to stable, causal, decaying exponentials, while poles
outside the unit circle correspond to anticausal exponentials that
decay toward time , and stop before time zero.

Figure 8.8 illustrates the two types of exponentials (causal and
anticausal) that correspond to poles (inside and outside the unit
circle) when the *z* transform region of convergence is defined to include the
unit circle.

myFourFiguresToWidthpolesout11polesout21polesout12polesout220.52Left column: Causal exponential decay, pole at . Right column: Anticausal exponential decay, pole at . Top: Pole-zero diagram. Bottom: Corresponding impulse response, assuming the region of convergence includes the unit circle in the plane.

**Next Section:**

Poles and Zeros of the Cepstrum

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Time Constant of One Pole